My Precalculus students have been asking me this question. I don’t really have a great answer, except that it’s true. Granted it’s not very intuitive. But nothing about infinite series is intuitive. For those not in my class or not familiar with the harmonic series, the question is:
And the answer is no. This is surprising, because the terms of the series approach zero.
Interestingly, the alternating harmonic series does converge:
And so does the p-harmonic series with p>1. For instance:
Besides looking at the sequence of partial sums, I’m not sure I can help you with the intuition on any of these facts. Can you?